If `A=[{:(p,q),(r,s):}]` satisfying equation `x^(2)-(p+s)x+lambda=0`, then

Options

A. `lambda=ps`
B. `lambda=qr`
C. `lambda=ps-qr`
D. `lambda=qr-ps`

Answer

To solve the problem, we need to analyze the given matrix \( A = \begin{pmatrix} p & q \\ r & s \end{pmatrix} \) and the quadratic equation \( x^2 - (p+s)x + \lambda = 0 \). We will find the relationship between \( \lambda \) and the elements of the matrix.

### Step-by-step Solution:

1. **Understanding the Matrix**:
We have a matrix \( A \) defined as:
\[ A = \begin{pmatrix} p & q \\ r & s \end{pmatrix} \]

2. **Calculating \( A^2 \)**:
We need to calculate \( A^2 \) using the matrix multiplication rule (Row-Column Rule):
\[ A^2 = A \cdot A = \begin{pmatrix} p & q \\ r & s \end{pmatrix} \cdot \begin{pmatrix} p & q \\ r & s \end{pmatrix} \]
Performing the multiplication:
\[ A^2 = \begin{pmatrix} p^2 + qr & pq + qs \\ rp + sr & rq + s^2 \end{pmatrix} \]

3. **Setting Up the Equation**:
We know that \( A^2 \) is related to the identity matrix \( I \) and \( \lambda \):
\[ A^2 - (p+s)A + \lambda I = 0 \]
This implies:
\[ A^2 = (p+s)A - \lambda I \]

4. **Substituting \( A \) and \( I \)**:
The identity matrix \( I \) in this case is:
\[ I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]
Thus, we can express \( \lambda I \) as:
\[ \lambda I = \begin{pmatrix} \lambda & 0 \\ 0 & \lambda \end{pmatrix} \]

5. **Equating the Matrices**:
We can equate the components of the matrices:
\[ \begin{pmatrix} p^2 + qr & pq + qs \\ rp + sr & rq + s^2 \end{pmatrix} = \begin{pmatrix} (p+s)p - \lambda & (p+s)q \\ (p+s)r & (p+s)s - \lambda \end{pmatrix} \]

6. **Finding \( \lambda \)**:
By comparing the elements of the matrices, we can derive the equations:
- From the first element:
\[ p^2 + qr = (p+s)p - \lambda \]
- From the second element:
\[ pq + qs = (p+s)q \]
- From the third element:
\[ rp + sr = (p+s)r \]
- From the fourth element:
\[ rq + s^2 = (p+s)s - \lambda \]

7. **Solving for \( \lambda \)**:
From the first and fourth equations, we can isolate \( \lambda \):
\[ \lambda = p^2 + qr - (p+s)p \]
\[ \lambda = (p+s)s - (rq + s^2) \]
Simplifying these gives us:
\[ \lambda = ps - qr \]

### Final Result:
Thus, the relationship between \( \lambda \) and the elements of the matrix is:
\[ \lambda = ps - qr \]